I'm icing an 11x15" sheet cake with buttercream icing, but I'm attaching swags along the sides. The swags will be made from gumpaste or fondant and hang down the sides. How do I attach the swags so that the weight doesn't pull them off the cake? I will also be attaching roses to the swags.

Melted chocolate or royal icing

is this a single layered cake? You called it a sheet cake so that makes me assume it's only 2" high, if that's the case you might consider sticking a toothpick in there to help hold it on.

It's going to be 2 layers, approx. 4" high.

Can I use royal icing to connect something to buttercream? I was under the impression that the grease in the buttercream would disrupt the adhesive properties of the royal icing?

grease disrupts the hardening properties of royal icing. The swags can be attached with buttercream icing. Add a small dollop or two to the curved portion as well as the top to help it adhere. When you add your roses (I'm assuming they'll be at the top of each swag?) use them to secure the tops of the swags.

I was going to leave my roses attached to toothpicks and put those through the swags. I figured those would adhere it better?

That's sounds like a winner!